by AMS - TEX

1991 Abstract. A topological space X is called A-real compact, if every algebra homomorphism from A to the reals is an evaluation at some point of X, where A is an algebra of continuous functions. Our main interest lies on algebras of smooth functions. In AdR] it was shown that any separable Banach space is smoothly real compact. Here we generalize this result to a huge class of locally convex spaces including arbitrary products of separable Fr echet spaces. In KMS] the notion of real compactness was generalized, by deen-ing a topological space X to be A-real-compact, if every algebra ho-momorphism : A ! R is just the evaluation at some point a 2 X, where A is a some subalgebra of C(X; R). In case A equals the algebra C(X; R) of all continuous functions this condition reduces to the usual real-compactness. Our main interest lies on algebras A of smooth functions. In particular we showed in KMS] that every space admitting A-partitions of unity is A-real-compact. Furthermore any product of the real line R is C 1-real-compact. A question we could not solve was, whether`1 is C 1-real-compact, despite the fact that there are no smooth bump functions. AdR] had already shown that this is true not only for`1 , but for any separable Banach space. The aim of this paper is to generalize this result of AdR] to a huge class of locally convex spaces, including arbitrary products of separable Fr echet spaces.